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- 1. Information Systems Society - Academics Committee Institute of Information and Computing Studies University of Santo Tomas By: J.G.M. Manuel MATH 101 (College Algebra) Reviewer for Preliminary Examinations MODULE 1 Real Numbers and their properties Real Numbers Rational Numbers – numbers which can be expressed as a ratio of 2 integers. Irrational Numbers – numbers which cannot be expressed as a ratio of 2 integers. Integers – consists of the set of whole numbers and their negative. Non-integers – Fractions, Non-terminating repeating Decimals, Terminating Decimals Whole Numbers – Counting numbers and Zero Natural Numbers – Counting numbers ONLY Properties of Real Numbers Commutative Property – you can add numbers in any order (Addition) Ex: 3a+2=2+3a - You can multiply numbers in any order (Multiplication) Ex: (3)(4)(6a)=(6a)(4)(3) Associative Property – you can group numbers in a sum in anyway you want and still get the same answer. (Addition) Ex.: (3x+2x)+9=3x+(2x+9) - You can group numbers in a product in anyway you want and still get the same answer. (Multiplication) Ex.:2x2(3y)=3y(2x2) Closure Property – The sum or product of real numbers is also a real number. Distributive Property (Distributive property of multiplication over addition) – If a term is multiplied by terms in a parenthesis, we need to distribute the term over all the terms inside the parenthesis. Identity Property – A number multiplied by 1 is the number itself. (Multiplication) - A number added to zero is the number itself. (Addition) Inverse Property – Any real number added/multiplied to its inverse is 0. Ex.: 3+(-3) = 0 (Addition) Ex.: 3(1/3)=0 For Order of Operations: P.E.M.D.A.S. – Parenthesis, Exponent, Multiply, Divide, Add, Subtract.
- 2. Information Systems Society - Academics Committee Institute of Information and Computing Studies University of Santo Tomas By: J.G.M. Manuel MODULE 2 Algebraic Expressions Algebraic Expression – combination of constants and variables related by the operations. Term - part of an algebraic expressions together with the plus or minus sign. Variable – symbol which stand for any element in a given set. It can be letters such as x,y and z. Constant - one possible value. Example: Algebraic Expression: ퟑ풙ퟐ+ ퟒ풙−ퟔ Terms: ퟑ풙ퟐ , 4x , -6 Factor – a term consists of a product of two or more constants or variables. Literal Factor – if the factor is a letter. Numerical Factor – if the factor is a number. Coefficient – the factor of a term of another factor. Example: Term: 7xyz Factors: 7, x, y, and z Numerical Factor: 7 Coefficient of yz: 7x Coefficient of xy: 7z Similar terms or Like terms – terms with same literal factors. Monomial – one term Example: ퟐ풙ퟐ풚ퟑ Binomial – two terms Example: 풙ퟐ− ퟐퟓ Trinomial – three terms Example: ퟓ풙ퟏ/ퟐ− ퟐ풙−ퟏ Multinomial – 2 or more terms Example: ퟑ풙ퟐ풚−ퟒ풙ퟑ+ ퟒ풚ퟐ− ퟐ
- 3. Information Systems Society - Academics Committee Institute of Information and Computing Studies University of Santo Tomas By: J.G.M. Manuel Polynomial – algebraic expressions involve constant or variables with positive integral exponents and no term with a variable in a denominator. Examples: Polynomials: ퟐ풙ퟐ− ퟔ풙+ퟖ ퟔ풚ퟐ+ ퟒ풚ퟑ풛+ퟐ ퟏퟒ풙ퟐ+ ퟏퟔ풙−ퟐ풙ퟑ Not Polynomials: ퟏퟎ풙ퟑ풚+ √풙+ ퟑ ퟔ풚ퟐ+ ퟐ풚ퟐ풛+ퟒ풛−ퟐ ퟕ풙ퟐ풚ퟐ+ ퟒ 풚 − ퟐ Degree of the polynomial – highest degree of any of the terms in a polynomial. Degree is the exponent of a variable. Example: ퟒ풙ퟑ풚ퟒ풛+ퟐ풙ퟐ풚ퟐ풛ퟐ− 풙풚 The degree with respect to x is 3. The degree with respect to xy is 2. The degree with respect to xyz is 8. Evaluating Algebraic Expressions -Replace the variables with the given value of variables, then perform the indicated operation. Example: Given: x=3, y=2, and z=-1 x3+y3+z3-2xyz 33+23+(-1)3-2(3)(2)(-1) 27+8-1+12 =46 Addition and Subtraction of Polynomials combining like terms 1. Identify terms with same literal coefficient. 2. Find the sum of each group of like terms by adding their numerical coefficient. Example: Subtract ퟐ풙ퟐ풚+ퟏퟐ−ퟓ풙ퟑ−ퟑ풙풚−ퟐ풙ퟐ풚ퟐ from the sum of −ퟒ풙ퟐ풚+ퟖ+ퟓ풙풚−ퟐ풙ퟑ− ퟐ풙ퟐ풚ퟐ and ퟕ−ퟑ풙ퟑ+ퟐ풙ퟐ풚+풙ퟐ풚ퟐ−ퟖ풙풚 =(−ퟒ풙ퟐ풚+ퟖ+ퟓ풙풚−ퟐ풙ퟑ−ퟐ풙ퟐ풚ퟐ)+ (ퟕ−ퟑ풙ퟑ+ퟐ풙ퟐ풚+풙ퟐ풚ퟐ−ퟖ풙풚) – (ퟐ풙ퟐ풚+ퟏퟐ−ퟓ풙ퟑ−ퟑ풙풚− ퟐ풙ퟐ풚ퟐ) =(−풙ퟐ풚ퟐ−ퟐ풙ퟐ풚−ퟑ풙풚−ퟓ풙ퟑ+ퟏퟓ) – (ퟐ풙ퟐ풚+ퟏퟐ−ퟓ풙ퟑ−ퟑ풙풚−ퟐ풙ퟐ풚ퟐ) =풙ퟐ풚ퟐ−ퟒ풙ퟐ풚+ퟑ Multiplication of Polynomials 1. Apply the distributive property, rules on exponent, and rules on signs. 2. Arrange the terms of polynomials in descending order.
- 4. Information Systems Society - Academics Committee Institute of Information and Computing Studies University of Santo Tomas By: J.G.M. Manuel Division of Polynomials -Arrange the terms descending order according to exponents. -Divide each term of the polynomial by the monomial using the given law of exponent -Simplify if possible. Removal of symbols of grouping Rules on removing symbols of grouping: 1. Removing a grouping symbol preceded by a (-) sign. Example: ퟓ풙– (ퟒ풚 – 풛) = ퟓ풙 – ퟒ풚 + 풛 2. Removing a grouping symbol preceded by a (+) sign. Example: ퟓ풙 + (ퟒ풚 – 풛) = ퟓ풙 + ퟒ풚 – 풛 3. Removing a grouping symbol preceded by a factor. Example: −ퟐ풙(ퟐ풙−ퟑ풙풚ퟐ)= −ퟒ풙ퟐ+ ퟔ풙ퟐ풚ퟐ ퟑ풚(ퟓ풙ퟐ풚−ퟐ풚ퟑ=ퟏퟓ풙ퟐ풚ퟑ−ퟐ풚ퟒ 4. Removing grouping symbols within other grouping symbol. Remove first the parenthesis () Then the braces [] The last is the {} Example: 풂{ퟐ풂풃 – ퟐ[풂(ퟐ풂풃 – 풃 + 풂풃) – ퟐ풂 (풂풃 + ퟐ) + 풂 ( 풃 – ퟑ풂)]} 풂{ퟐ풂풃 – ퟐ[ퟐ풂ퟐ풃−풂풃+ 풂ퟐ풃−ퟐ풂ퟐ풃−ퟒ풂+풂풃−ퟑ풂ퟐ]} 풂{ퟐ풂풃−ퟒ풂ퟐ풃+ퟐ풂풃−ퟐ풂ퟐ풃+ퟒ풂ퟐ풃−ퟖ풂−ퟐ풂풃+ퟔ풂ퟐ} ퟐ풂ퟐ풃−ퟒ풂ퟑ풃+ ퟐ풂ퟐ풃−ퟐ풂ퟑ풃+ퟒ풂ퟑ풃−ퟖ풂ퟐ−ퟐ풂ퟐ풃+ퟔ풂ퟑ ퟐ풂ퟐ풃−ퟐ풂ퟑ풃−ퟖ풂ퟐ+ퟔ풂ퟑ SPECIAL PRODUCTS Special Product is a multiplication of algebraic expression without using the long method. 1. Products of a Sum and Difference of Two Terms Form: (풂+풃)(풂−풃)=풂ퟐ−풃ퟐ Example: a) (ퟐ풙−ퟓ)(ퟐ풙+ퟓ)=ퟒ풙ퟐ−ퟐퟓ b) (ퟖ풙ퟑ−풚ퟒ)(ퟖ풙ퟑ+풚ퟒ)=ퟔퟒ풙ퟔ−풚ퟖ 2. Square of a Binomial Form: (풂±풃)ퟐ=(풂ퟐ±풂풃+풃ퟐ) Example: a) (ퟑ풙−ퟒ)ퟐ=(ퟑ풙)ퟐ−ퟐ(ퟑ풙)(ퟒ)+(ퟒ)ퟐ = ퟗ풙ퟐ−ퟐퟒ풙+ퟏퟔ b) (ퟗ풙ퟐ풚ퟑ+ퟒ풙ퟓ풚ퟕ) ퟐ =(ퟗ풙ퟐ풚ퟑ)ퟐ+ퟐ(ퟗ풙ퟐ풚ퟑ)(ퟒ풙ퟓ풚ퟕ)+(ퟒ풙ퟓ풚ퟕ) ퟐ =ퟖퟏ풙ퟐ풚ퟔ+ퟕퟐ풙ퟕ풚ퟏퟎ+ퟏퟔ풙ퟏퟎ풚ퟏퟒ
- 5. Information Systems Society - Academics Committee Institute of Information and Computing Studies University of Santo Tomas By: J.G.M. Manuel 3. Cube of a Binomial Form: (풂±풃)ퟑ=풂ퟑ±ퟑ풂ퟐ풃+ퟑ풂풃ퟐ±풃ퟑ Example: a) (ퟐ풙+ퟓ)ퟑ =(ퟐ풙)ퟑ+ퟑ(ퟐ풙)ퟐ(ퟓ)+ퟑ(ퟐ풙)(ퟓ)ퟐ+(ퟓ)ퟑ =ퟖ풙ퟑ+ퟔퟎ풙ퟐ+ퟏퟓퟎ풙+ퟏퟐퟓ 4. Product of Two Dissimilar Binomials Form: (풂+풃)(풄+풅)=풂풄+풂풅+풃풄+풃풅 FOIL method (first term-ac, outer term-ad, inner term-bc, last term-bd) Example: a) (풙−ퟒ)(풚+ퟐ) = 풙풚+ퟐ풙−ퟒ풚−ퟖ 5. Square of Polynomials Form: (풂+풃+풄)ퟐ=풂ퟐ+풃ퟐ+풄ퟐ+ퟐ풂풃+ퟐ풂풄+ퟐ풃풄 Example: a) (ퟐ풙−ퟑ풚+ퟒ풛)ퟐ =ퟒ풙ퟐ+ퟗ풚ퟐ+ퟏퟔ풛ퟐ−ퟏퟐ풙풚+ퟏퟔ풙풛−ퟐퟒ풚풛 FACTORING Factoring is the process of finding the factors. 1. Common Factor Form: 풂풃−풂풄+풂풅=풂(풃−풄+풅) Example: a) ퟑ풚ퟐ+ퟏퟐ풚 =ퟑ(풚ퟐ+ퟒ풚) →3 and 12 have a common factor of 3 =ퟑ풚(풚+ퟒ) →y2 and 4y have also a common factor which is y b) (풙+풚)ퟐ+풛(풙+풚) =(풙+풚)(풙+풚+풛) →(x+y)2 and z(x+y) have a common factor of (x+y) 2. Difference of two squares Form: 풂ퟐ−풃ퟐ=(풂+풃)(풂−풃) Example: a) 풙ퟐ−ퟏퟔ =(풙+ퟒ)(풙−ퟒ) →The √풙ퟐ is x and √ퟏퟔ is 4. b) ퟒ풙ퟐ−ퟗ =(ퟐ풙+ퟑ)(ퟐ풙−ퟑ) →get the square root of 4x2 and 9 3. Sum or difference of two cubes Form: 풂ퟑ±풃ퟑ=(풂±풃)(풂ퟐ∓풂풃+풃ퟐ) Example: a) 풙ퟑ−ퟖ =(풙)ퟑ−(ퟐ)ퟑ =(풙−ퟐ)(풙ퟐ+ퟐ풙+ퟐퟐ)
- 6. Information Systems Society - Academics Committee Institute of Information and Computing Studies University of Santo Tomas By: J.G.M. Manuel =(풙−ퟐ)(풙ퟐ+ퟐ풙+ퟒ) b) 풙ퟑ풚ퟔ+ퟔퟒ =(풙풚ퟐ)ퟑ−(ퟒ)ퟑ =(풙풚ퟐ+ퟒ)((풙풚ퟐ)ퟐ−(풙풚ퟐ)(ퟒ)+ퟒퟐ) =(풙풚ퟐ+ퟒ)(풙ퟐ풚ퟒ−ퟒ풙풚ퟐ+ퟏퟔ) 4. Sum or difference of odd powers Form: (풂풏±풃풏)=(풂±풃)(풂풏−ퟏ∓풂풏−ퟐ풃+풂풏−ퟑ풃ퟐ∓⋯±풃풏−ퟏ) Example: a) 풙ퟓ+ퟑퟐ =풙ퟓ+(ퟐ)ퟓ =(풙+ퟐ)(풙ퟒ−ퟐ풙ퟑ+(ퟐ)ퟐ풙ퟐ−(ퟐ)ퟑ풙+(ퟐ)ퟒ) =(풙+ퟐ)(풙ퟒ−ퟐ풙ퟑ+ퟒ풙ퟐ−ퟖ풙+ퟏퟔ) b) 풙ퟓ−ퟑퟐ =풙ퟓ−(ퟐ)ퟓ =(풙−ퟐ)(풙ퟒ+ퟐ풙ퟑ+(ퟐ)ퟐ풙ퟐ+(ퟐ)ퟑ풙+(ퟐ)ퟒ) =(풙−ퟐ)(풙ퟒ+ퟐ풙ퟑ+ퟒ풙ퟐ+ퟖ풙+ퟏퟔ) 5. Perfect square trinomial Form: 풂ퟐ±ퟐ풂풃+풃ퟐ=(풂±풃)ퟐ Example: a) 풙ퟐ−ퟏퟐ풙+ퟑퟔ →the first term, x2, is the square of x and the last term, 36, is =(풙−ퟔ)ퟐ the square of 6. Since the middle term has a “minus sign”, the 36 is the square of -6. b) ퟗ풙ퟐ+ퟐퟒ풙+ퟏퟔ =(ퟑ풙+ퟒ)ퟐ 6. Quadratic trinomial Form: 풂풄풙ퟐ+(풂풅+풃풄)풙+풃풅=(풂풙+풃)(풄풙+풅) Example: a) ퟓ풙ퟐ+ퟏퟏ풙+ퟐ →find the product of the first and last term: (5)(2)=10 =ퟓ풙ퟐ+ퟏ풙+ퟏퟎ풙+ퟐ →Think of two factors of 10 that adds up to 11: 1 and 10 =(ퟓ풙ퟐ+ퟏ풙)+(ퟏퟎ풙+ퟐ) →Group the two pairs of terms then remove common factors =풙(ퟓ풙+ퟏ)+ퟐ(ퟓ풙+ퟏ) →then we can factor out the common factor (5x+1) =(ퟓ풙+ퟏ)(풙+ퟐ) b) ퟒ풙ퟐ+ퟕ풙−ퟏퟓ →find the product of the first and last term: (4)(15)= - 60 =ퟒ풙ퟐ−ퟓ풙+ퟏퟐ풙−ퟏퟓ →Think of two factors of - 60 that adds up to 7: -5 and 12 =(ퟒ풙ퟐ−ퟓ풙)+(ퟏퟐ풙−ퟏퟓ) →Group the two pairs of terms then remove common factors =풙(ퟒ풙−ퟓ)+ퟑ(ퟒ풙−ퟓ) →then we can factor out the common factor (4x-5) =(ퟒ풙−ퟓ)(풙+ퟑ) 7. Factoring by adding or subtracting perfect square Example: Dapat ganito ang perfect square trinomial, kaya ang kulang niya ay 4x2
- 7. Information Systems Society - Academics Committee Institute of Information and Computing Studies University of Santo Tomas By: J.G.M. Manuel a) 풙ퟒ+ퟒ ↓ ↓ 풙ퟐ ퟐ풙ퟒ+ퟒ풙ퟐ+ퟒ =(풙ퟒ+ퟒ풙ퟐ+ퟒ)−ퟒ풙ퟐ →add and subtract 4x2 =(풙ퟐ+ퟐ)ퟐ−(ퟐ풙)ퟐ →then, factor it using the difference of two square =(풙ퟐ+ퟐ+ퟐ풙)(풙ퟐ+ퟐ−ퟐ풙) 8. Factoring by grouping Example: a) 풙ퟑ+ퟐ풙ퟐ+ퟖ풙+ퟏퟔ =풙ퟐ(풙+ퟐ)+ퟖ(풙+ퟐ) =(풙+ퟐ)(풙ퟐ+ퟖ) Rational Expression -is a fraction whose numerator and denominator are polynomials -can be simplified by applying factoring Equivalent Fraction – the numerator and denominator of a fraction may be multiplied or divided by a nonzero number without changing the value of the fraction. 풂 풃 = 풂풄 풃풄 풂 풃 = 풂 풄 풃 풄 Example: ퟑ풙 ퟑ(풙+ퟑ) = 풙 풙+ퟑ Simplifying Fractions 1. Factor all expression in numerator and denominator 2. Apply multiplicative cancellation Example: (풙ퟐ−ퟑퟔ) 풙−ퟔ = (풙+ퟔ)(풙−ퟔ) 풙−ퟔ =풙+ퟔ Multiplication and Division of Rational Expression 1. Factor all factorable algebraic expression on the given 2. Arrange the factors in the correct position, factor other factorable algebraic expression if possible. 3. Multiplicative cancellation, after cancellation combine all the expressions left. 풂 풃 ∙ 풄 풅 = 풂풄 풃풅 Example: (풙ퟐ−ퟐퟓ) ퟒ ∙ ퟐ풙ퟐ (푿−ퟓ) = (푿+ퟓ)(푿−ퟓ) ퟒ ∙ ퟐ풙ퟐ 푿−ퟓ = 풙ퟐ(푿+ퟓ) ퟐ There are no common factors to all four terms. So, we try grouping the first two together and the last two together, and factor out the common factor.
- 8. Information Systems Society - Academics Committee Institute of Information and Computing Studies University of Santo Tomas By: J.G.M. Manuel = 풙ퟑ+ퟓ풙ퟐ ퟐ Least Common Multiple (LCM) – the product of all the prime factors, with each raised to the highest power that occurs in any expression. Addition and Subtraction of Rational Expression 1. Factor completely all the denominators 2. Find the LCM for all denominators 3. Combine all the fractions to one single fraction 4. Distribute the numbers so that there would be no parenthesis, then combine like terms 5. Factor the remaining and apply multiplicative cancellation. Example: Complex Fraction - both numerator and denominator are fraction. 1. Search for the longest line, above it was the numerator and below it is the denominator. 2. Applying the rules in addition, subtraction, multiplication and division of rational expression, simplify the numerator and denominator to its simplest form. 3. Multiply the simplified form of numerator with the reciprocal of the denominator. RADICALS Radical is a number that has a fractional exponent √풙풏 Laws of Radicals 1. √풙풏∙√풚풏= √풙풚풏 Example: a) √ퟑ∙√ퟓ=√ퟏퟓ b) √ퟏퟒퟑ∙√ퟐퟑ= √ퟐퟖퟑ 2. √풙풏 √풚풏= √ 풙 풚 풏 Example: a) √ퟐퟎ ퟓ √ퟒ ퟓ=√ퟐퟎ ퟒ ퟓ =√ퟓퟓ b) √풙풚풛 √풂풃 =√ 풙풚풛 풂풃 3. √풙풎풏=(√풙풏) 풎 a) √풂ퟑퟒ=(√풂ퟒ)ퟑ 4. √√풙풏풎 = √풙풎∙풏 a) √√풂풃ퟒퟓ =√풂풃ퟐퟎ Simplifying Radicals 1. Radicand has no factor raised to a power greater than or equal to the index of radical. Example: a) √ퟐퟓퟔ풙ퟕ풚ퟒ풛ퟔퟑ =√ퟒퟒ풙ퟕ풚ퟒ풛ퟔퟑ =ퟒ풙ퟐ풚풛ퟐ√ퟒ풙풚ퟑ Index Radicand Radical sign
- 9. Information Systems Society - Academics Committee Institute of Information and Computing Studies University of Santo Tomas By: J.G.M. Manuel 2. No fraction appears on radicand. Example: a) √ 풂 풃풄 = √ 풂 풃풄 ∙ 풃풄 풃풄 = √풂풃풄 풃풄 3. No radicals within a radical Operations Involving Radicals Addition and Subtraction of Radicals → Terms with same index and radicands may be combined Example: a) √ퟑ−√ퟐퟕ+√ퟒퟖ =√ퟑ−√ퟑퟑ+√ퟒퟐ∙ퟑ =√ퟑ−ퟑ√ퟑ+ퟒ√ퟑ =ퟐ√ퟑ Multiplication of Radicals 1. Same Indices Ex. √ퟒ ∙ √ퟕ= √ퟒ∙ퟕ = √ퟐퟖ 2. Different Indices, but same Radicands Ex. √ퟒ∙ √ퟒퟑ = √ퟒퟐ+ퟑퟐ∙ퟑ = √ퟒퟓퟔ = √ퟏퟎퟐퟒퟔ 3. Different Indices and radicands Ex. √ퟑ∙ √ퟓퟑ=√ퟑퟑ∙ퟓퟐퟐ∙ퟑ = √ퟐퟕ∙ퟐퟓퟔ = √ퟔퟕퟓퟔ Division of Radicands - rewrite the expressions to fractional form and then use rationalization Ex. √ퟑퟐ풂ퟗ ퟐퟕ풃ퟐ ퟒ =√ퟐퟒ풂ퟖퟐ풂 ퟑퟑ풃ퟐ ퟒ =ퟐ풂ퟐ√ퟐ풂∙ퟐ풂 ퟑퟑ풃ퟐ∙ ퟑ풃ퟐ ퟑ풃ퟐ ퟒ =ퟐ풂ퟐ√ퟔ풂풃ퟐ ퟑퟒ풃ퟒ ퟒ = ퟐ풂ퟐ ퟑ풃 √ퟔ풂풃ퟐퟒ

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